a "problem" can be solved or can have the potential to be solved, while an "issue" has no specific 'right' answer. issue: 'whether to study chinese or not.' (debatable) problem: 'how to begin studying chinese?' (can be resolved with alternative solutions) :)

An "issue" is a matter that should be addressed or attended to. It is not necesarily a problem. But many times we use "issue" in place of "problem" so that the problem doesn't sound so serious or negative.

We sometimes use "issue" to imply a problem may not be a problem. In a manner of softening. ;) "He has a problem with controlling his drinking." --> "He has issues with controlling his drinking."

problem may refer to a state of difficulty that needs to be resolved and issue is an important question that is in dispute and must be settled cowboy...